As the demand for cloud computing services surges globally, so does its carbon footprint, making energy sustainability a critical challenge in the digital era. This paper explores the emerging field of carbon-aware cloud computing, which integrates carbon-intelligent decision-making into cloud infrastructure operations. We present a comprehensive survey of methodologies that incorporate real-time carbon intensity data, geographic workload shifting, and renewable energy-aware scheduling. Additionally, we examine the practical applications and system architectures that enable cloud providers to minimize environmental impact while maintaining performance and cost efficiency. Our findings highlight current advancements, identify gaps in existing strategies, and propose future research directions to drive the development of greener cloud computing ecosystems.